IN HYDRODYNAMICS. 43 



Ex. 6. Let it be required to determine whether in a compressible fluid the surfaces 

 of displacement can be spherical surfaces, the centres of which are always on the axis of .r, 

 and at the same time the motion be such that the radius of the surface of displacement of 

 a given particle remains the same in successive instants. 



Let {.V - af + ?/- + ^^ - R- = 0. Then by reasoning as in Ex. 3, F= f --J cos = K cos d. 



By hypothesis ( — ) is the same for all points of the same surface of displacement. Hence V^ 



is the velocity at any point of the axis of tv. The equation (9) may be put under the form, 



dp dV V dp 2V 

 pat as p as H 

 subject to the limitation of integrating along the line of motion s. The dynamical equation, 

 subject to the same limitation, is 



rdV V" 



k' Nap. log |o' + y — ds + — =/(0- 



Hence, carrying the approximation only to the first power of the velocity, we have 



^._ir, .nd'^=-f%ds^f(t). Therefore 

 pds dt pdt J at 



and differentiating with respect to s, 



(P^ ,P d-r ,0 f dF V dB\ 



^2 — 2 A; I I = 



dt- ' ds^ \Rds R"' dsl 



The motion is symmetrical about the axis of x, which is plainly a line of motion. Hence the 



above equation is true when F is put for Fand x for s. It thus becomes, 



d'V, „ d"-V „ f dV, V dR\ 



dt^ ' daf \Rdx R' ' dx) 



Now in this equation R = x - a, and = 1 - — , for a is a function both of x and t. The 



dx dx 



differential coefficient — will in all cases be very small, if the velocity of the particles be small 

 compared to the velocity of propagation of the motion. Hence -~ — = ^, I ' ~ w~ ) ~ ^* nearly, 



regarding — a quantity of the same order as F^. Also as F may be considered a function of 



„ , dF dV, dR dV, , . . d?V (PVdR tPF , „ u .% .• • 



R and t, '- = — — . = — - nearly. And '- = '-. ■ = —^ nearly. By substitution in 



dx dR dx dR ^ dx' dR' dx dR' ^ ^ 



the foregoing equation, we have 



- — • - A' . - — '- - afcM — '■ ■ =0. 



dt- dR' \RdR RV 



This equation gives F by integration, whence F is known from the equation F= F,cos0. Thus 



the motion is completely determined consistently with equation (9), and this is the proof of 



the possibility of the assumed kind of motion, so far, at least, as regards small motions. The 



above solution is that whicli I have employed for finding the resistance of the air to the vibrations 



of a ball-pendulum. 



CAMBBinOB ObSERVA'TORV. 



Mardi 2, ]84.'3. 



